If I beat the game on casual, and select to restart the game with my existing stuff after beating the final boss, then change the gameplay difficulty right after I start, would I get the trophy for beating it on hard after that run through?

A regular playthrough has to be completed in the set difficulty to unlock the trophy (since another separate trophy covers the Challenge Mode playthrough). If you finish Hard with one or both previous difficulty trophies missing, you will get them at the same time.